7 EFFECTIVE
RADON DIFFUSION COEFFICIENT
7.1 DEFINITION
The random movement of the radon gas atoms mixed in the air results
in a net migration of the radon gas toward the direction of its decreasing
concentration in the air. This phenomenon is called molecular or atom diffusion.
The diffusion of radon in open air can be described by Fick's law, which
states that the flux density of the diffusing substance is linearly proportional
to its concentration gradient. Fick's law can be expressed as follows:
where J is a vector representing the density flux of radon activity
in units of activityl-2T-1, 
is a vector representing the gradient of radon activity concentration in
the air in units of activityl-4, and Do is
the molecular (or atom) diffusivity or the diffusion coefficient of radon
in open air in units of l2T-1. Therefore, the diffusion
coefficient Do can be defined from Fick's equation and
expressed as the ratio of the magnitudes of the vectors J to 
:
For radon diffusion in open air, Fick's law is uniquely expressed and,
consequently, the diffusion coefficient of radon in open air, Do,
is also uniquely defined. However, when applied to the conditions of radon
diffusion in porous media, such as in soil materials, Fick's equation can
be written in different ways, depending on how the variables flux density
J and concentration C are defined. Fick's equation can be
written in four distinct ways when applied to the molecular diffusion phenomenon
in porous media, depending on whether the bulk or pore volume is used to
define the concentration and whether the bulk or pore area is used to define
the flux density. These different ways of defining the radon diffusion
coefficient in soil lead to some confusion in selecting and using these
parameters because the symbols and nomenclature used have not been standardized
(Nazaroff et al. 1988).
Two distinct ways of defining the diffusion coefficient of radon in
porous media have been adopted in the literature: (1) De
is the effective radon diffusion coefficient and (2) D is the
bulk radon diffusion coefficient. However, Culot (1976) and Nazaroff et
al. (1988) have noted discrepancies regarding the way these two coefficients
are defined and used in modeling the diffusion of radon through porous
media. Therefore, the definitions of De and D
adopted in this handbook are those suggested by Nazaroff et al. (1988).
Thus, the effective (or interstitial) radon diffusion coefficient, De,
is defined from Fick's equation as the ratio of the diffusive flux density
of radon activity across the pore area, Je, to the gradient
of the radon activity concentration in the pore or interstitial space,
. This definition
is equivalent to that relating the bulk flux density to the gradient of
the bulk concentration of radon activity in the soil and can be expressed
as follows:
The bulk radon diffusion coefficient, D, is defined as the ratio
of the diffusive flux density of radon activity across a geometric or superficial
area of the medium, Jb, to the gradient of the radon
activity concentration in the pore space, 
,
and can be expressed as follows:
The bulk and the effective radon diffusion coefficients in soil, D
and De, respectively, are correlated by the total soil
porosity, pt, according to the following expression:
In general, the diffusion coefficient in porous media is a property
of the diffusing species, the pore structure, the type of fluids present
in the pores, the adsorption properties of the solid matrix, the fluid
saturations, and temperature. For radon diffusion in porous media, the
diffusivity for the other isotopes of radon (e.g., radon-220) has been
observed to be comparable to that for the isotope radon-222 (Nazaroff et
al. 1988).
Several attempts have been made to correlate the radon diffusion coefficients
in porous media (D and De) to the radon diffusion
coefficient in open air (Do) and the physical properties
of the medium such as the total porosity (pt). These
attempts have not been conclusive. According to experimental work performed
by Currie (1960a,b) and quoted by Rolston (1986) and Nazaroff et al. (1988),
the coefficients D and Do can be correlated by
an expression of the following form:
where and µ represent measures of pore shape of the soil materials.
This empirical relationship can fit data from a wide range of dry porous
materials in which the values of generally lie between 0.8 and 1.0 and
the values of µ lie around 1.0. This empirical relationship is not
applicable, however, for very wet soil and strongly aggregate soil (Rolston
1986).
The influence of soil moisture content on the effective diffusion coefficient
of radon in soil has been investigated by Rogers and Nielson (1991), who
proposed the following expression:
where Do = 1.1 × 10-5 m2/s
is the radon diffusivity in open air, pt is the total
soil porosity, and Rs is the water saturation in the
soil (or the fraction of the pore space filled with water, also called
the saturation ratio).
7.2 MEASUREMENT METHODOLOGY
The diffusivity (or the diffusion coefficient) of radon in soils can
be measured by both field and laboratory experiments. In either case, the
experimental evaluation of the diffusivity consists in determining the
numerical value of the coefficient appearing in Fick's equation. Because
of the difficulties in implementing field methods, laboratory methods are
generally used to determine the radon diffusivity in porous media and particularly
in soil materials.
Variations of the laboratory methods for measuring radon diffusivity
in porous media have been developed and as yet no standard (or recommended)
method exists. All the various laboratory methods are based on the solution
of the mass balance equation that represents the diffusion process in a
one-dimensional configuration. Depending on the approximation taken on
the time domain for the solution of the diffusion equation, these methods
can be separated into two distinct groups: (1) the steady-state diffusion
method and (2) the transient diffusion method (Nielson et al. 1982).
The steady-state method used in the laboratory for the determination
of the radon diffusivity in soil material without a source of radon within
it is based on the solution of a one-dimensional diffusion equation in
the x-direction, expressed as follows:
This steady-state equation is obtained by coupling the one-dimensional
Fick's equation,
with the one-dimensional, steady-state, continuity equation,
where Je is the effective flux density of radon activity
(pCi)/(m2s), C is the concentration of radon activity
in the pore space (pCi/m3), and is the radon decay constant
(1/s).
A steady-state diffusion method for determining the effective radon
diffusion coefficient (De) in uncontaminated (no radon
source) soil materials was implemented by Silker and Kalkwarf (Silker 1981;
Silker and Kalkwarf 1983) on the basis of theoretical developments by Cohen
(1979). The apparatus used in this method consists of a column of test
soil of known depth, d, which is sealed at one end to an air chamber
of known volume containing a radon source with a known and constant strength.
The other end of the test soil column is kept open. As a boundary condition
for this system, it is assumed that in a steady-state situation, the effective
flux density of radon activity at the bottom of the column, Jeo,
is constant and uniquely dependent on the strength of the radon source
and the geometry of the system. Also, the radon activity concentration
at the open end of the soil column is assumed to be negligible (i.e., zero).
On the basis of these assumptions and conditions, the effective radon
diffusivity, De, can then be evaluated by the following
equation (Silker and Kalkwarf 1983):
where Co is the radon activity concentration within
the air chamber, and l is the radon diffusion length (or relaxation
length) parameter within the porous medium, which is defined as follows:
The right side of Equation 7.12 is a well-defined function of the parameter
ratio d/l and is independent of the measured values of Co
and Jeo. The left side of the equation is dependent
on the measured results. Therefore, by selecting the size (i.e., thickness)
of the soil test sample, d; determining the effective flux density
Jeo on the basis of the strength of the radon source and
the column diameter; and making several measurements of Co; Equation 7.9
can be graphically or numerically solved for the ratio d/l and subsequently
for De.
Typically, the soil samples used in the determination of De
have a cylindrical shape with a height to 10 cm and an inner diameter of
14 cm. After equilibration, the steady-state radon concentration in the
bottom chamber, Co, is determined by several measurements
taken over a 7- to 14-day period. Each measurement consists of withdrawing
about 5 cm3 of gas from a typical 800-cm3 bottom
chamber and determining the radon concentration by using either a scintillation
flask technique (such as a Lucas cell) or charcoal absorption and gamma-ray
spectrometry (Silker 1983).
7.3 RESRAD DATA INPUT REQUIREMENTS
In RESRAD, the user is requested to input an effective diffusion coefficient
value of radon for three materials: (1) the soil of the cover zone,
(2) the soil of the contaminated zone, and (3) the building foundation
material (i.e., concrete). The dimensions of these input values of De
are in units of square meters per second (m2/s). For each
porous material considered, the value of De is assumed
to be the same for both radon isotopes addressed in RESRAD, that is, radon-222
and radon-220.
The effective radon diffusivity values in porous media (soils and concrete
included) vary over a wide range of several orders of magnitude depending
on the porous material and particularly on its degree of water saturation.
Table 7.1 lists representative values of effective diffusion coefficients
of radon obtained by different researchers for a range of unconsolidated
soil materials, concrete, and other building materials. Because of the
differences in the experimental methodologies adopted by the various researchers,
these experimental data are not easily comparable. Nevertheless, they may
give an indication of the expected values of De in the
field.
Typically, the effective diffusion coefficient of radon in unconsolidated
soil material with a low moisture content is about 10-6 m2/s.
The upper limit is represented by the radon diffusion coefficient in open
air, Do, which is about 1.1 × 10-5
m2/s. At the lower extreme, in a fully saturated soil material
the radon diffusion coefficient may be as low as 10-10 m2/s.
In RESRAD, a default value of De equal to 2.0 × 10-6
m-2/s was adopted for both the cover and contaminated zones.
According to the data presented in Table 7.1, this default value of
De would represent the average effective radon diffusion
coefficient in soils with a lower moisture content and composed of silty
and clayey sands. The observed range of variation of De in
concrete, as presented in Table 7.1, goes from 8.0 × 10-9
to 4.0 × 10-7 m2/s. A default value
of De equal to 3.0 × 10-7 m2/s
was adopted in the RESRAD model to represent the effective radon diffusion
coefficient in concrete.
The estimation of the values of the effective radon diffusion coefficient (De) to be used in RESRAD can be performed at different levels of site-specific accuracy, depending on the amount of information available. For generic use of the code, a set of default values of De was defined as 2.0 × 10-6 m2/s for the cover and contaminated zones and 3.0 × 10-7 m2/s for the building foundation (i.e., concrete). If the type of soil materials at the site is known, a
| TABLE 7.1 Effective Diffusion
Coefficients for Radon in Unconsolidated Soil Materials
and Concretea |
|||
Porous Material |
Effective Radon Diffusion Coefficient, De (m2s-1) |
Comment |
Reference |
Unconsolidated soil material Compacted silty sands Compacted clayey sands Compacted inorganic clays Silty sandy clay Uranium mill tailings Loams Mud Concrete Other materials Brick Gypsum |
(3.0±1.3) × 10-6 (3.2±1.5) × 10-6 (2.5±1.0) × 10-6 2.7 × 10-6 2.5 × 10-7 6.0 × 10-8 (5.4-7.2) × 10-6 8 × 10-7 5.7 × 10-10 (1.1-4.0) × 10-7 1.2 × 10-8 3.4 × 10-8 3.3 × 10-8 (0.8-8.4) × 10-8 (0.8-3.0) × 10-7 (1.0-4.0) × 10-6 |
pt = 0.29-0.36 Rs = 0.05-0.34 pt = 0.32-0.39 Rs = 0.09-0.55 pt = 0.32-0.43 Rs = 0.06-0.34 w = 1.5% dry weight w = 10.5% dry weight w = 17.3% dry weight w = (0.7-1.5)% dry weight Dry = 37% pt = 0.11-0.13 pt = 0.25 pt = 0.05 pt = 0.068 - - - |
Silker and Kalkwarf (1983) Silker and Kalkwarf (1983) Silker and Kalkwarf (1983) Strong et al. (1981) Strong et al. (1981) Tanner (1964) Tanner (1964) Poffijn et al. (1988) Culot et al. (1976) Culot et al. (1976) Zapalac (1983) Stranden (1988) Stranden (1988) Stranden (1988) |
a pt = total porosity, Rs = volumetric water saturation, w = percent water content by weight, and = percent volumetric water content. Source: Adapted from Nazaroff et al. (1988). |
|||
slightly more accurate estimation of De can be performed
with the help of Table 7.1. For most applications, this approach will suffice
because of the natural variability of De within the soil
and building materials of any specific site.
In cases in which there are reasons to suspect that the default values
of the effective radon diffusion coefficient (De) do
not reflect the conditions at a specific site and there is no possibility
of measuring De, the RESRAD code is able to estimate
it internally on the basis of the values of the water saturation (calculated
from the volumetric water content) and total porosity, according to Equation
7.7. To implement this option, the user should enter any negative number
as an input value of De to RESRAD.
For an accurate site-specific estimate of the input data to RESRAD,
however, the values of De should be measured in either
the laboratory or field experiments. Whenever necessary and possible, the
measurement of De in the soil cover zone (it is assumed
that it is not contaminated with radon sources) should be performed in
the laboratory by using a method such as the Silker and Kalkwarf (1983)
technique.
8.1 DEFINITION
The radon emanation coefficient, , is the fraction of the total amount
of radon produced by radium decay that escapes from the soil particles
and gets into the pores of the medium. It is also called the emanating
power, emanating fraction, release ratio, and escape-to-production ratio.
The radon emanation coefficient is a dimensionless parameter and is represented
as either a fraction or a percentage.
The two most common radioisotopes of radon gas, radon-222 and radon-220,
are generated by a radioactive process of alpha decay from two radium isotopes,
radium-226 and radium-224, respectively. Because of the conservation of
linear momentum in the alpha-decay process, the newly created radon-222
and radon-220 atoms are left with a kinetic (usually called "recoil")
energy of about 86 and 103 keV, respectively (Nazaroff et al. 1988).
Thus, after being generated, the radon atoms tend to move away from
their original location until their recoil energy is totally transferred
to the medium. Consequently, depending on their original location within
the solid phase of the soil, the soil pore distribution, and the soil moisture
content, the newly created radon atoms may end up within the same soil
particle in which they were created, within the adjacent soil particle
because of posterior penetration after escaping from the host soil grain,
or within the pore of the medium.
Experimental data reported by several investigators indicate that the
radon emanation coefficient is strongly influenced by the moisture content
of the medium, particularly within the range of low water saturation (Nazaroff
et al. 1988). On the basis of results of this kind, it has been hypothesized
that the amount of water present in the pore increases the absorption of
the recoil energy of the radon atom passing through it, thus enhancing
the chance that the atom will terminate its recoil within the water. Partition
equilibrium of radon in the water and air phases in the pore will follow
afterwards based on Henry's law.
Although temperature may influence the magnitude of the radon emanation
coefficient, it has been demonstrated that within the normal range of temperature
variability of surface soils, this effect is of minor importance (Nazaroff
et al. 1988).
The radon emanation coefficient, , is one of the characteristic soil parameters that determine the rate of radon emanation into the pores of the soil matrix. The other soil characteristic parameter in relation to radon production is the concentration of radium (radium-226 and/or radium-224) in the soil particles, SRa. In RESRAD, the source of radon
generation in the pore air or the rate of radon generation and emanation
into the soil gas phase (pore air), S., in units of pCi/m3s,
is calculated as follows:
where is the radon emanation coefficient (dimensionless), s
is the soil particle density (kg/m3), SRa
is the mass concentration of radium (radium-226 or radium-224(4))
in the soil particles (pCi/kg), is the radon (radon-222 or radon-220) decay
constant (1/s), and pt is the total porosity of the contaminated
soil.
The values of the radon emanation coefficient in soils depend on the
radon isotope being considered, the soil material, and the moisture content.
Experimental measurements of in different soils, rocks, and other materials
have been reported by many investigators. Table 8.1 presents a summary
of these available data. Because of the differences in the experimental
methodologies adopted by the various investigators, these data are not
easily comparable. The data are also incomplete in that they do not reflect
a rigorous and systematic analysis of the radon emanation coefficient for
all radon isotopes in a broad range of soil materials and rocks under different
degrees of water saturation. Although incomplete, these available data
may give an indication of the expected values of in the field.
8.2 MEASUREMENT METHODOLOGY
The methodology for measuring the radon emanation coefficient () of
a porous material contaminated with radium consists basically of measuring
the radon concentration in the air within a sealed accumulation chamber
in which a sample of the contaminated soil material has been left for a
period of time (around four days) until the radon concentration reaches
equilibrium. A detailed description of a variation of this method is presented
in Strong and Levins (1982). Their experimental apparatus consisted of
an ingrowth (accumulation) chamber, a sampling cylinder, a diaphragm pump,
a scintillation cell, and supporting electronics for the radiation measurement.
8.3 RESRAD DATA INPUT REQUIREMENTS
In RESRAD, the user is requested to input a value for the radon emanation coefficient () that is related to the soil material of the contaminated zone for the two radon isotopes, radon-222 and radon-220. This parameter is dimensionless and its value should be entered as a fraction (rather than as a percentage).
| TABLE 8.1 Measurements of Emanation
Coefficients of Radon (Rn-222 and Rn-220) in Unconsolidated
Soils and Other Materials |
|||||
Material |
Number of Samples |
Isotope |
Emanation Coefficienta |
Moisture Content |
Reference |
Unconsolidated soils Sand Sandy loam Silty loam (Heavy) loam Clay Various soils (Danish) Soil Soil Other materials Uranium ore (crushed) Uranium mill tailings |
7 7 7 12 5 70 21 2 17 2 |
Rn-222 Rn-222 Rn-222 Rn-222 Rn-222 Rn-222 Rn-222 Rn-220 Rn-222 Rn-222 |
0.14 (0.06 - 0.18) 0.21 (0.10 - 0.36) 0.24 (0.18 - 0.40) 0.20 (0.17 - 0.23) 0.28 (0.18 - 0.40) 0.22 (0.02 - 0.70) 0.30 (0.03 - 0.55) 0.12 (0.09 - 0.15) 0.28 (0.06 - 0.55) 0.14 (0.02 - 0.36) (0.29 - 0.31) (0.067 - 0.072) |
Unknown Unknown Unknown Unknown Unknown 0-70% dry wt Unknown Oven-dried Moist, saturated Vacuum-dried Saturated Oven-dried |
Sisigina (1974) Sisigina (1974) Sisigina (1974) Sisigina (1974) Sisigina (1974) Damkjaer and Korsbech (1985) Barreto (1974) Megumi and Mamuro (1974) Thamer et al. (1981) Thamer et al. (1981) Strong and Levins (1982) Strong and Levins (1982) |
a Arithmetic mean (range of values). Source: Adapted from Nazaroff et al. (1988). |
|||||
As shown in Table 8.1, the radon emanation coefficient varies from 0.02
to 0.70 in soils. The values of for radon-222 are usually higher than those
for radon-220 under the same circumstances. In RESRAD, the adopted default
values of the radon emanation coefficient () for radon isotopes radon-222
and radon-220 are, respectively, 0.25 and 0.15, in the soil of the contaminated
zone. These default values approximately represent the conditions in a
silty loam soil with a low moisture content (i.e., not dry).
The estimation of the values of for radon-222 and radon-220 for use
in RESRAD can be performed at different levels of site-specific accuracy,
depending on the amount of information available. For generic use of the
code, a set of default values for (0.25 for radon-222 and 0.15 for radon-220)
was defined to approximately represent the condition of a silty loam soil
with a low moisture content in the contaminated zone.
If the type of soil materials at the site is known, a slightly more
accurate estimation of can be performed with the help of the data in Table
8.1. For most applications, this approach will suffice because of the natural
variability of within the soil of the contaminated zone of any specific
site.
In cases in which it is absolutely necessary to have an accurate estimate
of and there are reasons to suspect that the data in Table 8.1 do not reflect
the conditions at a specific site, the values of for radon-222 and radon-220
can be determined experimentally in the laboratory by using the previously
mentioned method of Strong and Levins (1982).